3.880 \(\int \frac{1}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx\)

Optimal. Leaf size=255 \[ \frac{3 b \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac{3 b \log (x)}{a^4}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c x^2}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*x^2) + (b^2 - 2*a*c + b*
c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)^2) + (3*b^4 - 20*a*b^2*c + 20*
a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*x^2*(a + b*x^2 + c*x^4
)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x^2)/S
qrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[x])/a^4 + (3*b*Log[a +
b*x^2 + c*x^4])/(4*a^4)

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Rubi [A]  time = 0.894886, antiderivative size = 255, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444 \[ \frac{3 b \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac{3 b \log (x)}{a^4}-\frac{3 \left (b^2-5 a c\right ) \left (b^2-2 a c\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac{20 a^2 c^2+3 b c x^2 \left (b^2-6 a c\right )-20 a b^2 c+3 b^4}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}-\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}+\frac{-2 a c+b^2+b c x^2}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^2 + c*x^4)^3),x]

[Out]

(-3*(b^2 - 5*a*c)*(b^2 - 2*a*c))/(2*a^3*(b^2 - 4*a*c)^2*x^2) + (b^2 - 2*a*c + b*
c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)^2) + (3*b^4 - 20*a*b^2*c + 20*
a^2*c^2 + 3*b*c*(b^2 - 6*a*c)*x^2)/(4*a^2*(b^2 - 4*a*c)^2*x^2*(a + b*x^2 + c*x^4
)) - (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*ArcTanh[(b + 2*c*x^2)/S
qrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)) - (3*b*Log[x])/a^4 + (3*b*Log[a +
b*x^2 + c*x^4])/(4*a^4)

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Rubi in Sympy [A]  time = 138.41, size = 258, normalized size = 1.01 \[ \frac{- 2 a c + b^{2} + b c x^{2}}{4 a x^{2} \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} + \frac{20 a^{2} c^{2} - 20 a b^{2} c + 3 b^{4} + 3 b c x^{2} \left (- 6 a c + b^{2}\right )}{4 a^{2} x^{2} \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} - \frac{3 \left (- 5 a c + b^{2}\right ) \left (- 2 a c + b^{2}\right )}{2 a^{3} x^{2} \left (- 4 a c + b^{2}\right )^{2}} - \frac{3 b \log{\left (x^{2} \right )}}{2 a^{4}} + \frac{3 b \log{\left (a + b x^{2} + c x^{4} \right )}}{4 a^{4}} - \frac{3 \left (- 20 a^{3} c^{3} + 30 a^{2} b^{2} c^{2} - 10 a b^{4} c + b^{6}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{2 a^{4} \left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(c*x**4+b*x**2+a)**3,x)

[Out]

(-2*a*c + b**2 + b*c*x**2)/(4*a*x**2*(-4*a*c + b**2)*(a + b*x**2 + c*x**4)**2) +
(20*a**2*c**2 - 20*a*b**2*c + 3*b**4 + 3*b*c*x**2*(-6*a*c + b**2))/(4*a**2*x**2
*(-4*a*c + b**2)**2*(a + b*x**2 + c*x**4)) - 3*(-5*a*c + b**2)*(-2*a*c + b**2)/(
2*a**3*x**2*(-4*a*c + b**2)**2) - 3*b*log(x**2)/(2*a**4) + 3*b*log(a + b*x**2 +
c*x**4)/(4*a**4) - 3*(-20*a**3*c**3 + 30*a**2*b**2*c**2 - 10*a*b**4*c + b**6)*at
anh((b + 2*c*x**2)/sqrt(-4*a*c + b**2))/(2*a**4*(-4*a*c + b**2)**(5/2))

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Mathematica [A]  time = 1.12735, size = 402, normalized size = 1.58 \[ \frac{\frac{a^2 \left (-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right )}{\left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}-\frac{a \left (46 a^2 b c^2+28 a^2 c^3 x^2-29 a b^3 c-26 a b^2 c^2 x^2+4 b^5+4 b^4 c x^2\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{3 \left (-20 a^3 c^3+30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}-10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}+b^6\right ) \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{3 \left (20 a^3 c^3-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt{b^2-4 a c}+10 a b^4 c+b^5 \sqrt{b^2-4 a c}-8 a b^3 c \sqrt{b^2-4 a c}-b^6\right ) \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}-\frac{2 a}{x^2}-12 b \log (x)}{4 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^2 + c*x^4)^3),x]

[Out]

((-2*a)/x^2 + (a^2*(b^3 - 3*a*b*c + b^2*c*x^2 - 2*a*c^2*x^2))/((-b^2 + 4*a*c)*(a
+ b*x^2 + c*x^4)^2) - (a*(4*b^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x^2 - 26*
a*b^2*c^2*x^2 + 28*a^2*c^3*x^2))/((b^2 - 4*a*c)^2*(a + b*x^2 + c*x^4)) - 12*b*Lo
g[x] + (3*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c
] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b - Sqrt[b
^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2) + (3*(-b^6 + 10*a*b^4*c - 30*a^2*b^2
*c^2 + 20*a^3*c^3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2
*b*c^2*Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5
/2))/(4*a^4)

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Maple [B]  time = 0.038, size = 1486, normalized size = 5.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(c*x^4+b*x^2+a)^3,x)

[Out]

-7/a/(c*x^4+b*x^2+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6+13/2/a^2/(c*x^4+b*x^2+
a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*b^2-1/a^3/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*
c^2-8*a*b^2*c+b^4)*x^6*b^4-37/2/a/(c*x^4+b*x^2+a)^2*b*c^3/(16*a^2*c^2-8*a*b^2*c+
b^4)*x^4+55/4/a^2/(c*x^4+b*x^2+a)^2*b^3*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-2/a^3
/(c*x^4+b*x^2+a)^2*b^5*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4-9/(c*x^4+b*x^2+a)^2/(16*
a^2*c^2-8*a*b^2*c+b^4)*x^2*c^3-7/2/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4
)*x^2*b^2*c^2+6/a^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^4*c-1/a^3
/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*b^6-29/2/(c*x^4+b*x^2+a)^2*b/(
16*a^2*c^2-8*a*b^2*c+b^4)*c^2+9/a/(c*x^4+b*x^2+a)^2*b^3/(16*a^2*c^2-8*a*b^2*c+b^
4)*c-5/4/a^2/(c*x^4+b*x^2+a)^2*b^5/(16*a^2*c^2-8*a*b^2*c+b^4)+12/a^2/(16*a^2*c^2
-8*a*b^2*c+b^4)*c^2*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*b-6/a^3/(16*a
^2*c^2-8*a*b^2*c+b^4)*c*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*b^3+3/4/a
^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln((16*a^2*c^2-8*a*b^2*c+b^4)*(c*x^4+b*x^2+a))*b^5
-30/a/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-
b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+b^4
)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b
^10)^(1/2))*c^3+45/a^2/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^
6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*
c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6
*c^2+20*a*b^8*c-b^10)^(1/2))*b^2*c^2-15/a^3/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a
^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*c*x^2*(16*a^2*c^2-8*
a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-1280*a^4*b^2*c^4+640*a^
3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b^4*c+3/2/a^4/(1024*a^5*c^5-12
80*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2)*arctan((2*
c*x^2*(16*a^2*c^2-8*a*b^2*c+b^4)+(16*a^2*c^2-8*a*b^2*c+b^4)*b)/(1024*a^5*c^5-128
0*a^4*b^2*c^4+640*a^3*b^4*c^3-160*a^2*b^6*c^2+20*a*b^8*c-b^10)^(1/2))*b^6-1/2/a^
3/x^2-3*b*ln(x)/a^4

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^3*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.965477, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^3*x^3),x, algorithm="fricas")

[Out]

[-1/4*(3*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 - 20*a^3*c^5)*x^10 + 2*(b^7*c
- 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^8 + (b^8 - 8*a*b^6*c + 10*a^2
*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^6 + 2*(a*b^7 - 10*a^2*b^5*c + 30*a^3*b
^3*c^2 - 20*a^4*b*c^3)*x^4 + (a^2*b^6 - 10*a^3*b^4*c + 30*a^4*b^2*c^2 - 20*a^5*c
^3)*x^2)*log((b^3 - 4*a*b*c + 2*(b^2*c - 4*a*c^2)*x^2 + (2*c^2*x^4 + 2*b*c*x^2 +
b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c*x^4 + b*x^2 + a)) + (6*(a*b^4*c^2 - 7*a^2*b^
2*c^3 + 10*a^3*c^4)*x^8 + 2*a^3*b^4 - 16*a^4*b^2*c + 32*a^5*c^2 + 3*(4*a*b^5*c -
29*a^2*b^3*c^2 + 46*a^3*b*c^3)*x^6 + 2*(3*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2
+ 50*a^4*c^3)*x^4 + (9*a^2*b^5 - 68*a^3*b^3*c + 122*a^4*b*c^2)*x^2 - 3*((b^5*c^2
- 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^10 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*x
^8 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^6 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*
c^2)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2)*log(c*x^4 + b*x^2 + a) +
12*((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^10 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^
2*b^2*c^3)*x^8 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^6 + 2*(a*b^6 - 8*a^2*b^4*c +
16*a^3*b^2*c^2)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*x^2)*log(x))*sqrt(
b^2 - 4*a*c))/(((a^4*b^4*c^2 - 8*a^5*b^2*c^3 + 16*a^6*c^4)*x^10 + 2*(a^4*b^5*c -
8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^8 + (a^4*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*x^6 +
2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*x^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c
^2)*x^2)*sqrt(b^2 - 4*a*c)), 1/4*(6*((b^6*c^2 - 10*a*b^4*c^3 + 30*a^2*b^2*c^4 -
20*a^3*c^5)*x^10 + 2*(b^7*c - 10*a*b^5*c^2 + 30*a^2*b^3*c^3 - 20*a^3*b*c^4)*x^8
+ (b^8 - 8*a*b^6*c + 10*a^2*b^4*c^2 + 40*a^3*b^2*c^3 - 40*a^4*c^4)*x^6 + 2*(a*b^
7 - 10*a^2*b^5*c + 30*a^3*b^3*c^2 - 20*a^4*b*c^3)*x^4 + (a^2*b^6 - 10*a^3*b^4*c
+ 30*a^4*b^2*c^2 - 20*a^5*c^3)*x^2)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^
2 - 4*a*c)) - (6*(a*b^4*c^2 - 7*a^2*b^2*c^3 + 10*a^3*c^4)*x^8 + 2*a^3*b^4 - 16*a
^4*b^2*c + 32*a^5*c^2 + 3*(4*a*b^5*c - 29*a^2*b^3*c^2 + 46*a^3*b*c^3)*x^6 + 2*(3
*a*b^6 - 18*a^2*b^4*c + 7*a^3*b^2*c^2 + 50*a^4*c^3)*x^4 + (9*a^2*b^5 - 68*a^3*b^
3*c + 122*a^4*b*c^2)*x^2 - 3*((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*x^10 + 2*(b
^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*x^8 + (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*x^6
+ 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*x^4 + (a^2*b^5 - 8*a^3*b^3*c + 16*a^4
*b*c^2)*x^2)*log(c*x^4 + b*x^2 + a) + 12*((b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)
*x^10 + 2*(b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*x^8 + (b^7 - 6*a*b^5*c + 32*a^3
*b*c^3)*x^6 + 2*(a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*x^4 + (a^2*b^5 - 8*a^3*b^
3*c + 16*a^4*b*c^2)*x^2)*log(x))*sqrt(-b^2 + 4*a*c))/(((a^4*b^4*c^2 - 8*a^5*b^2*
c^3 + 16*a^6*c^4)*x^10 + 2*(a^4*b^5*c - 8*a^5*b^3*c^2 + 16*a^6*b*c^3)*x^8 + (a^4
*b^6 - 6*a^5*b^4*c + 32*a^7*c^3)*x^6 + 2*(a^5*b^5 - 8*a^6*b^3*c + 16*a^7*b*c^2)*
x^4 + (a^6*b^4 - 8*a^7*b^2*c + 16*a^8*c^2)*x^2)*sqrt(-b^2 + 4*a*c))]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(c*x**4+b*x**2+a)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 15.6525, size = 516, normalized size = 2.02 \[ \frac{3 \,{\left (b^{6} - 10 \, a b^{4} c + 30 \, a^{2} b^{2} c^{2} - 20 \, a^{3} c^{3}\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{9 \, b^{5} c^{2} x^{8} - 72 \, a b^{3} c^{3} x^{8} + 144 \, a^{2} b c^{4} x^{8} + 18 \, b^{6} c x^{6} - 136 \, a b^{4} c^{2} x^{6} + 236 \, a^{2} b^{2} c^{3} x^{6} + 56 \, a^{3} c^{4} x^{6} + 9 \, b^{7} x^{4} - 38 \, a b^{5} c x^{4} - 110 \, a^{2} b^{3} c^{2} x^{4} + 436 \, a^{3} b c^{3} x^{4} + 26 \, a b^{6} x^{2} - 192 \, a^{2} b^{4} c x^{2} + 316 \, a^{3} b^{2} c^{2} x^{2} + 72 \, a^{4} c^{3} x^{2} + 19 \, a^{2} b^{5} - 144 \, a^{3} b^{3} c + 260 \, a^{4} b c^{2}}{8 \,{\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )}{\left (c x^{4} + b x^{2} + a\right )}^{2}} + \frac{3 \, b{\rm ln}\left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac{3 \, b{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} + \frac{3 \, b x^{2} - a}{2 \, a^{4} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2 + a)^3*x^3),x, algorithm="giac")

[Out]

3/2*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3)*arctan((2*c*x^2 + b)/sqrt(-
b^2 + 4*a*c))/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) - 1/8*(9
*b^5*c^2*x^8 - 72*a*b^3*c^3*x^8 + 144*a^2*b*c^4*x^8 + 18*b^6*c*x^6 - 136*a*b^4*c
^2*x^6 + 236*a^2*b^2*c^3*x^6 + 56*a^3*c^4*x^6 + 9*b^7*x^4 - 38*a*b^5*c*x^4 - 110
*a^2*b^3*c^2*x^4 + 436*a^3*b*c^3*x^4 + 26*a*b^6*x^2 - 192*a^2*b^4*c*x^2 + 316*a^
3*b^2*c^2*x^2 + 72*a^4*c^3*x^2 + 19*a^2*b^5 - 144*a^3*b^3*c + 260*a^4*b*c^2)/((a
^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c*x^4 + b*x^2 + a)^2) + 3/4*b*ln(c*x^4 + b*x
^2 + a)/a^4 - 3/2*b*ln(x^2)/a^4 + 1/2*(3*b*x^2 - a)/(a^4*x^2)